Estimates similar to those in Theorem REF  hold for a broad class of generalized projections (see [1]} and the references therein), and recently Shmerkin [2]} proved an analogue of Theorem REF  where the set of orthogonal projections \(\lbrace \pi _e\rbrace _{e\in S^1}\)  is replaced by a family \(\lbrace F_{\lambda }\rbrace _{\lambda \in \Lambda }\)  of nonsingular \(C^2\)  functions \(F_\lambda \colon \mathbb {R}^2\rightarrow \mathbb {R}\) , referred to as nonlinear projections, that satisfy a natural transversality condition.
